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Research Papers

# Distributed Operational Space Formulation of Serial Manipulators

[+] Author and Article Information
Kishor D. Bhalerao

Software Engineer
Immersive Technologies,
Perth, WA 6017, Australia;
Honorary Research Fellow,
University of Melbourne,
Melbourne, VIC 3010, Australia
e-mail: kishorb@unimelb.edu.au

James Critchley

Multibody.org,
Lake Orion, MI 48362
e-mail: James@multibody.org

Denny Oetomo

Senior Lecturer
Department of Mechanical Engineering,
University of Melbourne,
Melbourne, VIC 3010, Australia
e-mail: doetomo@unimelb.edu.au

Roy Featherstone

Consultant
Istituto Italiano di Tecnologia,
Genova 16163, Italy
e-mail: roy.featherstone@ieee.org

Oussama Khatib

Professor
Department of Computer Science,
Stanford University,
Stanford, CA 94305
e-mail: ok@cs.stanford.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 2, 2013; final manuscript received September 9, 2013; published online October 30, 2013. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 9(2), 021012 (Oct 30, 2013) (10 pages) Paper No: CND-13-1118; doi: 10.1115/1.4025577 History: Received June 02, 2013; Revised September 09, 2013

## Abstract

This paper presents a new parallel algorithm for the operational space dynamics of unconstrained serial manipulators, which outperforms contemporary sequential and parallel algorithms in the presence of two or more processors. The method employs a hybrid divide and conquer algorithm (DCA) multibody methodology which brings together the best features of the DCA and fast sequential techniques. The method achieves a logarithmic time complexity ($O(log(n)$) in the number of degrees of freedom ($n$) for computing the operational space inertia ($Λe$) of a serial manipulator in presence of $O(n)$ processors. The paper also addresses the efficient sequential and parallel computation of the dynamically consistent generalized inverse ($J¯e$) of the task Jacobian, the associated null space projection matrix ($Ne$), and the joint actuator forces ($τnull$) which only affect the manipulator posture. The sequential algorithms for computing $J¯e$, $Ne$, and $τnull$ are of $O(n)$, $O(n2)$, and $O(n)$ computational complexity, respectively, while the corresponding parallel algorithms are of $O(log(n))$, $O(n)$, and $O(log(n))$ time complexity in the presence of $O(n)$ processors.

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Topics: Algorithms

## Figures

Fig. 1

The DCA formalism

Fig. 2

Notation for a serial manipulator

Fig. 3

Divide and conquer algorithm

Fig. 4

Time comparison for computing Je and Jex

Fig. 5

Time comparison for computing Λe-1 and βe

Fig. 6

Time comparison for computing Λe-1 and βe

Fig. 7

Load balancing for computing Λe-1 and βe for the 512 body chain

Fig. 8

Time comparison for computing Λe-1 and βe

Fig. 9

Time comparison for computing J¯e

Fig. 10

Time comparison for computing Ne

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